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Manuscript Title: CheMPS2: improved DMRG-SCF routine and correlation functions
Authors: Sebastian Wouters, Ward Poelmans, Stijn De Baerdemacker, Paul W. Ayers, Dimitri Van Neck
Program title: CheMPS2
Catalogue identifier: AESE_v2_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 191(2015)235
Programming language: C++.
Computer: x86-64.
Operating system: Linux.
RAM: 10MB - 512GB
Keywords: SU(2) spin-adapted DMRG, ab initio quantum chemistry, DIIS, Edmiston-Ruedenberg orbital localization, Fiedler vector, two-orbital mutual information.
PACS: 31.15.ae, 31.15.vj, 31.50.Df.
Classification: 16.1.

External routines: Basic Linear Algebra Subprograms (BLAS), Linear Algebra Package (LAPACK), GNU Scientific Library (GSL), Hierarchical Data Format Release 5 (HDF5), and Open Multi-Processing (OpenMP)

Does the new version supersede the previous version?: Yes

Nature of problem:
The many-body Hilbert space grows exponentially with the number of single-particle states. Exact diagonalization solvers can therefore only handle small active spaces, of up to 16 electrons in 16 orbitals. Interesting active spaces are often significantly larger.

Solution method:
The density matrix renormalization group allows to extend the size of active spaces, for which numerically exact solutions can be found, significantly. In addition, it provides a rigorous variational upper bound to energies, as it has an underlying wavefunction ansatz, the matrix product state.

Reasons for new version:
The DMRG routine is 20% faster in the new version. Several features were added to the augmented Hessian Newton-Raphson DMRG-SCF routine. In addition, five correlation functions were implemented to study the electronic structure in a comprehensible way. A python interface to the library is provided.

Summary of revisions:
  • The DMRG routine is 20% faster compared to the previous version [1]. The active space of N2 in the cc-pVDZ basis consists of 14 electrons in 28 orbitals. We have calculated the X1Σg+ ground state near equilibrium. Canonical orbitals are used (D2h symmetry), grouped in irrep blocks, and sorted to place bonding and anti-bonding blocks adjacent [1]. The average wall time per sweep on a dual Intel Xeon Sandy Bridge E5-2670 (total of 16 cores at 2.6 GHz) is shown in Table 1 for the new and previous program versions, for several values of the reduced virtual dimension DSU(2).

    Table 1: The average wall time per sweep on a dual Intel Xeon Sandy Bridge E5-2670 (total of 16 cores at 2.6 GHz) for the X1Σg+ ground state of N2 near equilibrium in the cc-pVDZ basis. Canonical orbitals are used, grouped in irrep blocks, and sorted to place bonding and antibonding blocks adjacent. The new version is 20% faster than the previous version.

  • DSU(2)   tnew [s]   tprevious [s]   tnew / tprevious  
    1000 117 152 0.77
    1500 279 356 0.78
    2000 529 668 0.79
    3000 1339 1647 0.81

  • Several features have been added to the augmented Hessian Newton-Raphson DMRG-SCF routine. An instance of the DMRGSCFoptions class has to be passed to CASSCF::doCASSCFnewtonraphson, which allows to overwrite the default options:
    • DMRGSCFoptions::setDoDIIS(bool value) : If value is true, a speed-up of the augmented Hessian Newton-Raphson DMRG-SCF routine is achieved with the direct inversion of the iterative subspace (DIIS) [2]. Examples can be found in test6, test8 and test9.
    • DMRGSCFoptions::setWhichActiveSpace(int value) : If value is 0, no rotations are performed in addition to the DMRG-SCF occupied-active, active-virtual, and occupied-virtual rotations. If value is 1, natural orbitals are used in the active space, sorted within each irrep according to their occupation number. If value is 2, localized and ordered orbitals are used in the active space. The localization is performed by maximizing the Edmiston-Ruedenberg cost function [3]. Thereto we have implemented a Newton-Raphson maximization algorithm, with exact Hessian of the cost function. The ordering is determined by approximately minimizing the bandwidth of the exchange matrix with the Fiedler vector [4]. Examples can be found in test6, test8 and test9.
    • DMRGSCFoptions::setStateAveraging(bool value) : When an excited state is targeted with DMRG-SCF and value is true, the DMRG-SCF gradient and Hessian are calculated based on an equally weighted average of the 2-RDMs of the targeted state and all lower-lying states. An example can be found in test6.
  • The electronic structure can be analyzed by means of the two-orbital mutual information [5,6]

    I(i,j) = 1/2 (S1(i) + S1(j) - S2(i,j)) (1 - δij) ≥ 0,              (1)

    where S1(i) is the single-orbital entropy of orbital i and S2(i,j) the two-orbital entropy of orbitals i and j. Additional information is contained in the following correlation functions:
    • The spin correlation function
      Cspin(i,j) = 4 〈ŜizŜjz〉 - 4 〈Ŝiz〉 〈Ŝjz
    • The spin-flip correlation function
      Cspinflip(i,j) = 〈Ŝi+Ŝj-〉 + 〈Ŝi-Ŝj+
    • The density correlation function
      Cdens(i,j) = 〈 ninj〉 - 〈ni〉 〈 nj
    • The singlet diradical correlation function [7]
      Cdirad(i,j) = 〈di↑dj↓〉 + 〈di↓dj↑〉 - 〈di↑〉 〈dj↓〉 - 〈di↓〉 〈dj↑,
      where d = n(1 - ni - σ)

    After a DMRG calculation, these five correlation functions can be printed by calling DMRG::calc2DMandCorrelations() and subsequently DMRG::getCorrelations()->Print(). An example can be found in test4. During DMRG-SCF calculations, they can be printed by setting DMRGSCFoptions::setDumpCorrelations(true). An example can be found in test8.

  • Restrictions:
    Our implementation of the density matrix renormalization group is spin-adapted. This means that targeted eigen-states in the active space are exact eigenstates of the total electronic spin operator. Hamiltonians which break this symmetry cannot be handled by our code.

    Unusual features:
    The nature of the matrix product state ansatz allows for exact spin coupling. In CheMPS2, the total electronic spin is imposed (not just the spin projection), in addition to the particle-number and abelian point-group symmetries.

    Additional comments:
    A more elaborate overview of the new features can be found in the CASSCF, Correlations, DMRGSCFunitary and EdmistonRuedenberg class references in the doxygen documentation, either generated by the instructions in README.md or online [8]. Updated versions of CheMPS2 will be provided at its public git repository [9].

    Running time:
    Examples are given in Table 1. It should be mentioned that the running time depends strongly on the size of the targeted active space, the density of states, the orbital choice and ordering, and the reduced virtual dimension DSU(2).

    [1] S. Wouters, W. Poelmans, P. W. Ayers and D. Van Neck, CheMPS2: A free open-source spin-adapted implementation of the density matrix renormalization group for ab initio quantum chemistry, Computer Physics Communications 185 (6), 1501-1514 (2014). http://dx.doi.org/10.1016/j.cpc.2014.01.019
    [2] T. Yanai, Y. Kurashige, D. Ghosh and G. K.-L. Chan, Accelerating convergence in iterative solution for large-scale complete active space self-consistent-field calculations, International Journal of Quantum Chemistry 109 (10), 2178-2190 (2009). http://dx.doi.org/10.1002/qua.22099
    [3] C. Edmiston and K. Ruedenberg, Localized atomic and molecular orbitals, Reviews of Modern Physics 35 (3), 457-464 (1963). http://dx.doi.org/10.1103/RevModPhys.35.457
    [4] G. Barcza, O. Legeza, K. H. Marti and M. Reiher, Quantum-information analysis of electronic states of different molecular structures, Physical Review A 83 (1), 012508 (2011). http://dx.doi.org/10.1103/PhysRevA.83.012508
    [5] J. Rissler, R. M. Noack and S. R. White, Measuring orbital interaction using quantum information theory, Chemical Physics 323 (2-3), 519-531 (2006). http://dx.doi.org/10.1016/j.chemphys.2005.10.018
    [6] G. Barcza, R. M. Noack, J. Solyom and O. Legeza, Entanglement patterns and generalized correlation functions in quantum many body systems (2014). http://arxiv.org/abs/1406.6643
    [7] J. Hachmann, J. J. Dorando, M. Aviles and G. K.-L. Chan, The radical character of the acenes: A density matrix renormalization group study, Journal of Chemical Physics 127 (13), 134309 (2007). http://dx.doi.org/10.1063/1.2768362
    [8] S. Wouters, CheMPS2 documentation, http://sebwouters.github.io/CheMPS2/index.html (2014).
    [9] S. Wouters, CheMPS2: a spin-adapted implementation of DMRG for ab initio quantum chemistry, https://github.com/SebWouters/CheMPS2 (2014).